Experimental investigation and thermodynamic correlation of chlordiazepoxide solubility in supercritical CO₂

Abstract

This study investigated the solubility of chlordiazepoxide in supercritical carbon dioxide (SCCO₂) at pressures ranging from 12 to 30 (MPa) and temperatures ranging from 308 to 338 (K). The measured solubilities ranged from 19.8 to 576 parts per million (ppm), or mole fractions of 0.0198 × 10⁻³ to 0.576 × 10⁻³. We used two equations of state (SRK and ECM-PR), enlarged liquid models, and semi-empirical analysis to assess the experimental data. Applying these models to the experimental solubility data of the chlordiazepoxide-SCCO₂ system revealed that Chrastil’s empirical models best aligned with the two sets of data. An important step in understanding the system’s behavior was applying the SRK and UNIQUAC models to the drug’s solubility data. Density-dependent empirical models, particularly the Chrastil correlation (AARD% = 5.30, R² = 0.996) and the UNIQUAC model (AARD% = 6.12), provided the highest predictive accuracy for chlordiazepoxide solubility in SCCO₂. In contrast, cubic EoS models (SRK and ECM-PR) exhibited significant deviations, with AARD% values approaching 20%, suggesting their limited reliability for this system. The UNIQUAC model indicated excellent accuracy, with AARD% of 6.12 and R² of 0.981 in correlating solubility data of this binary system.

Introduction

When it comes to high-pressure methods, supercritical fluid (SCF) technology is at the forefront. that offers significant advantages in enhancing product performance across various industries. SCFs are environmentally friendly, and their promising economic potential has led to its growing commercial adoption in sectors such as pharmaceuticals, textiles, and food industry^1,2,3. Drug delivery encompasses a diverse array of formulations and methodologies designed to efficiently manage pharmaceutical compounds to achieve specific therapeutic objectives^4,5,6. Conventional methods of producing drug delivery systems have certain limitations, which calls for new and improved strategies. to overcome these limitations. Traditional methods like jet milling, spray drying, hot-melt extrusion, and solvent casting, face challenges such as high residual solvent contents, and difficulties in controlling particle size and distribution^7,8. These issues can lead to physical and chemical instability in the final product, restrictions in applicability to temperature-sensitive compounds, and concerns regarding the thermal and chemical stability of pharmaceutical ingredients^7,8. Given these challenges, there is a growing interest in the pharmaceutical industry in exploring alternative methods, such as supercritical processes, to develop efficient drug delivery systems. By leveraging the benefits of SCF technology, researchers are aiming at addressing the shortcomings of conventional techniques and enhancing the formulation and delivery of pharmaceutical products^7,8. As a solvent, SCCO₂ has many benefits, including allowing for exact regulation of particle size, reduced solvent consumption, and improved stability of drug formulations. SCCO₂ is widely recognized as the most used SCF in various industries, particularly in the pharmaceutical one. Its popularity stems from several key advantages, including its FDA-approved status as a solvent, cost-effectiveness, non-toxic properties, recyclability, moderate critical conditions, and easy accessibility. These characteristics make SCCO₂ a preferred choice for pharmaceutical applications, as it offers a safe and environmentally-friendly solvent option that meets regulatory standards while providing efficient and effective solvation properties for drug delivery systems^{7,8,9,10,11,12,13}. SCF have shown significant potential in the production of micro and nanoparticles, offering improved solubility and controlled particle engineering, as demonstrated in recent studies. The versatility of SCCO₂ was demonstrated in several ways, including the formation of nano- or microparticles from pigments and dyes^11,12,14,15. the extraction of essential oils and seed oils^2,9,16,17,18, formation of particles^19,20,21,22, and impregnation processes^{23,24,25,26,27,28,29}. Indeed, numerous SCCO₂-based techniques have been established for formulating efficient drug delivery systems, such as pharmaceutical micro- and nanoparticles and diverse drug-polymer combinations. These methods encompass loading fine pharmaceutical particles onto a polymer surface, encapsulating them within the polymer matrix, or co-precipitating them with polymer particles.

The challenge posed by the limited solubility of polar and complex pharmaceutical compounds in SCCO₂ is a significant hurdle in the development of supercritical pharmaceutical processes, particularly when CO₂ is used as a solvent. This limitation arises from the nonpolar nature of CO₂ molecules, which hinders their ability to establish specific interactions with polar solute molecules^30,31,32,33. The addition of polar co-solvents to SCCO₂ has been shown to enhance its solvation capability and improve the supercritical solubility of pharmaceutical compounds³¹. However, in the present work, no co-solvent such as ethanol was used. The solubility behavior of chlordiazepoxide was investigated in a binary system with pure SCCO₂ to evaluate its intrinsic solubility profile and interaction with the solvent. Supercritical solubility tests in a lab are useful, but they can be costly and time-consuming. Consequently, several theoretical methods, including models based on empirical density, models based on EoS, and expanded-liquid models, have been explored to efficiently analyze the supercritical solubility of compounds. Identifying a reliable predictive model for supercritical solubility requires a thorough comparison of different models against the relevant laboratory data sets. Therefore, the optimal correlation model can vary depending on the specific solute^{34,35,36,37,38}. The simplicity and reliability of empirical models make them the go to choose for connecting supercritical solubility data. reasonable accuracy within the range of available data, and independence from thermophysical properties that are typically challenging to estimate for pharmaceutical substances. However, they are generally not suitable for extrapolation or predictive purposes beyond the experimental conditions^39,40,41. These models have shown effectiveness in accurately predicting solubility behavior in supercritical conditions. Certain researchers have indeed treated SCCO₂ as a condensed gas in their models and have used fugacity coefficients determined by applying the relevant EoSs to the solute. This approach has been used to develop models that aim at accurately predicting solubility behavior in supercritical conditions. The EoSs can be categorized into two main groups: cubic equations on one hand, such as the Soave EoS on one side, we have non-cubic equations grounded in statistical associating fluid theory (SAFT); on the other, we have EoSs that are written as cubic functions of molar volume, such as Peng Robinson (PR) and Soave-Redlich-Kwong (SRK). Indeed, some researchers have considered SCF to be an expanded form of liquid due to their high density, which closely resembles that of liquids. Expanded-liquid models treat SCCO₂ as an effectively highly expanded liquid and apply liquid-phase thermodynamic concepts to predict solute solubility. Two broad classes are commonly used: (i) density-dependent empirical correlations (e.g., Chrastil, Bartle, MST), which correlate solubility directly with solvent density and temperature and require only a few fitted parameters; and (ii) activity-coefficient–based approaches (e.g., expanded-liquid UNIQUAC, Modified Wilson and NRTL), which use excess Gibbs-energy models adapted for supercritical conditions to capture nonideal solute–solvent interactions. Density-dependent correlations are simple and often highly accurate within the fitted range, while expanded-liquid UNIQUAC, NRTL and Modified Wilson provide a more physically grounded description for systems with specific interactions but require binary interaction parameters. In this study we apply both types to assess predictive performance for chlordiazepoxide in SCCO₂^36,42,43,44. Thermodynamic modeling is influenced by the specific temperature and pressure conditions of a process, along with physical and chemical properties of the pharmaceutical compound, some of which (e.g., critical properties) must be estimated.

When trying to correlate results from different experiments, models are vital. The sempiternal dichotomy about solubility models is whether to use empirical or EoS models. Although empirical ones may be more accurate for a given set of experimental data, they are not convenient when extrapolation is needed. In that case, extrapolation is more reliable when using an EoS model.

Chlordiazepoxide is an FDA-approved drug indicated for the treatment of mild to severe anxiety disorders, preoperative anxiety and apprehension, as well as managing withdrawal symptoms associated with acute alcohol use disorder in adults. Additionally, it is approved for use in pediatric patients aged six years and older for anxiety management. This research investigates the solubility behavior of the drug in SCCO₂ across a pressure range of 12–30 MPa and temperatures between 308 and 348 K. Furthermore, the supercritical system’s chlordiazepoxide solubility was correlated using established empirical models and the SRK and ECM-PR model based on EoS and the UNIQUAC model.

To clearly highlight the contribution of this study, it should be noted that, to the best of our knowledge, no experimental solubility data for chlordiazepoxide in pure SCCO₂ have been reported in the literature. Therefore, the present work provides the first equilibrium solubility dataset for this pharmaceutical compound under supercritical conditions (pressure range of 12–30 MPa and temperatures between 308 and 348 K). This new dataset enables a systematic evaluation of several thermodynamic models; including empirical correlations, the SRK and ECM-PR equations of state, and the UNIQUAC expanded-liquid model, for a drug for which no prior modeling has been performed.

Laboratory section

Materials

High-purity compounds; chlordiazepoxide (used as a solute), carbon dioxide (used as a solvent), and ethanol were employed in this study. You can find more specific details on these compounds in Table 1. Figure 1 depicts the chlordiazepoxide molecular structure.

Table 1 Chemicals used in this work.

Fig. 1

Structure of chlordiazepoxide.

Equipment and experimental procedure

Lenalidomide¹, fludrocortisone acetate⁴⁵, and montelukast³³ were among the drugs whose solubilities have been determined using the experimental equipment shown in Fig. 2. A comprehensive explanation of the solubility measurement procedure and relevant calculation equations is available in references^1,26,27. A brief summary of the procedure is as follows (see Fig. 2):

(1) CO₂ gas is first filtered using a 1 μm molecular sieve, (2) Then cooled and liquefied in a refrigerator, (3) Pressurized to target levels (12–30 MPa) using a high-pressure pump (± 0.1 MPa precision), (4) Preheated to the desired temperature (308–338 K) and introduced into an oven-controlled equilibrium cell (± 1 K), (5) In order to achieve phase equilibrium, the chlordiazepoxide powder (2 g) and CO₂ should be mixed in the equilibrium cell using a magnetic stirrer for approximately 180 min. This will increase the mass transfer between the phases. (6) The injection loop is used to transfer some of the saturated solution to a vial containing methanol after reducing its pressure (7). Then, at 269 nm, the UV-Vis spectrophotometry (Perkin-Elmer) is used to quantify the quantity of dissolved chlordiazepoxide.

Fig. 2

Experimental setup for solubility measurements.

Modeling

The solubility of chlordiazepoxide in SCCO₂ was correlated in the present study using multiple models. These include empirical models (Sect. "Empirical models"), EoS models (Sect. "Equation of state-based model"), and the expanded liquid theory (Sect. "Expanded liquid model (UNIQUAC)").

Empirical models

This study tested the feasibility of four empirical models for predicting chlordiazepoxide solubility in SCCO₂. A total of three constants can be adjusted across all four models. Tabulated in Table 2 are the model and their equations. As a function of pressure (MPa), temperature (K), and SCCO₂ density (kg/m³), these empirical models illustrate how a solute dissolve in supercritical CO₂. Using standard statistical measures such as R-squared, and AARD% (average absolute relative deviation), we assessed how well the models describe chlordiazepoxide solubility data in this binary system performed. These measures are useful for gauging the model’s fit to the experimental data and for understanding the precision of the predictions. The equations used to compute these parameters have been previously reported in the literature¹.

Table 2 Empirical solubility models used in this work.

Equation of state-based model

To establish the equilibrium condition between supercritical CO₂ (as the solvent) and chlordiazepoxide (as the solute), thermodynamic equilibrium criteria between the two phases are applied. An equation of state is utilized to facilitate this analysis. A drug’s solubility in a supercritical fluid can be expressed as follows since the solute’s fugacity is identical in both phases at equilibrium:

$${y_2}=\frac{{P_{2}^{{sub}}}}{P}\,\frac{{\varphi _{2}^{{sat,s}}}}{{{\varphi _2}}}\,\exp \left[ {\frac{{v_{2}^{s}\left( {P - P_{2}^{{sub}}} \right)}}{{R\,T}}} \right]$$

(1)

To constructing this equation, it was presumed that the solute does not contain SCCO₂, that the solid solute is free of impurities, and that the incompressible solid’s density (and, by extension, its molar volume) remains constant regardless of pressure. At a given pressure (P) and temperature (T), the solute’s equilibrium mole fraction (y₂) can be determined using Eq. (1). In this equation, $P_{2}^{{sub}}$ represents the sublimation pressure, $v_{2}^{s}$ the molar volume of the solute, and ${\varphi _2}$ the fugacity coefficient of the solute. The saturation fugacity coefficient of the solute $\varphi _{2}^{{sat,s}}$ is evaluated at the sublimation pressure at which it can be safely assumed to be equal to 1⁴⁶. Additionally, ${\varphi _2}$ can be determined using an appropriate equation of state³⁰ as follows:

$$RT\ln {\varphi _2}=\int\limits_{V}^{\infty } {\,\left[ {{{\left( {\frac{{\partial P}}{{\partial {n_2}}}} \right)}_{T,V,{n_{j \ne {n_2}}}}} - \frac{{RT}}{V}} \right]\,} dV - RT\ln \frac{{PV}}{{nRT}}$$

(2)

Where, n₂ represents the moles of solute, V is the volume.

Soave–Redlich–Kwong (SRK) model

Because of its acceptable accuracy in modeling the solubility in SCF, as validated by various researchers^1,47, the Soave equation of state (SRK) has been chosen to evaluate the partial derivative in Eq. (2), which requires an equation of state. SRK can be express it is:

$$P=\frac{{RT}}{{v - b}} - \frac{{a(T)}}{{v\,\,(v+b)}}$$

(3)

Parameter a(T) is obtained using the following relationships:

$$a(T)=\frac{{0.42747{R^2}T_{c}^{2}}}{{{P_c}}}\,\alpha ({T_{r,\omega }})$$

(4)

where:

$$\alpha ({T_{r,\omega }})={[1+m(1 - T_{r}^{{0.5}})]^2}$$

(4a)

and:

$$m=0.480+1.574\omega - 0.176{\omega ^2}$$

(4b)

Parameter b is calculated by:

$$b=\frac{{0.08664R{T_c}}}{{{P_c}}}$$

(5)

The experimental data was modeled using the Soave EoS, which is a variant of the Redlich-Kwong EoS and is often called Soave-Redlich-Kwong EoS (SRK). Parameters am and bm for the Soave EoS in the binary system chlordiazepoxide + SCCO₂ were determined using conventional van der Waals mixing procedures. These are given by⁴⁸:

$${a_m}=\sum\nolimits_{j} {{y_i}} {y_j}\,\sqrt {{a_i}{a_j}} \left( {1 - {k_{ij}}} \right)$$

(6)

$${b_m}=\sum\nolimits_{j} {{y_i}} {y_j}\,\frac{{{b_i}+{b_j}}}{2}\left( {1 - {l_{ij}}} \right)$$

(7)

with l_ij and k_ij serving as the so-called criteria for interaction.

The necessary properties of chlordiazepoxide were estimated using group contribution methods. Prior studies have employed these techniques to determine the physicochemical properties of various drug molecules^30,45. Table 3 displays the required properties of chlordiazepoxide for solubility calculations as well as the method used to estimate them.

Table 3 Chlordiazepoxide properties obtained by group contribution methods.

In order to determine a solid’s solubility in SCF, the sublimation pressures of the solute are necessary for any EoS approach. Experimental values of sublimation pressures are scarce due to the difficulties in measuring them and, therefore, they must be normally estimated. In this case, the Ambrose-Walton corresponding states method⁴⁶ was used. Table 4 shows the estimated values needed in this case. These uncertainties can influence solubility predictions by affecting fugacity and sublimation pressure calculations. While model fitting reduces reliance on absolute accuracy of input properties, the impact of these estimation errors on solubility calculations should be considered in sensitivity analyses.

Table 4 Chlordiazepoxide sublimation pressures.

A drug’s physicochemical qualities, especially its sublimation pressure, have a significant impact on its solubility in supercritical CO2. (P^sub), enthalpy of sublimation (ΔH^sub), and molecular structure. Sublimation pressure determines the volatility of the solute and directly affects the equilibrium between solid and vapor phases. A higher sublimation pressure at a given temperature corresponds to higher solubility in SCFs^46,52. Additionally, ΔH^sub plays a critical role; a lower ΔH^sub facilitates the transition of molecules from the solid phase into the supercritical fluid^46,53. Furthermore, the molecular structure, including polarity, hydrogen bonding ability, and molar volume, affects solvation interactions with CO₂^30,42,54. These properties influence key parameters in both EoS-based models (e.g., fugacity coefficients) and activity coefficient models such as UNIQUAC^40,42. Therefore, accurate estimation or measurement of sublimation-related parameters is essential for predictive and mechanistic modeling of solubility in SCFs. It is worth noting that chlordiazepoxide has a high melting point (514.75 K), well above the temperature range used in this study. Although CO₂ can depress the melting point of certain substances, the operating conditions here were not sufficient to induce melting. Hence, the drug remained in solid form and the measured solubilities reflect solid–supercritical fluid (solid–vapor) equilibrium.

ECM-PR model

The EoS model proposed by Estévez et al.⁵⁵. will be used here. The interesting feature of this model, as compared to other EoS ones, is that it does not require the critical properties of the solute since they are embedded into two adjustable parameters α and β. This approach can be used with any EoS. The original publication⁵⁵ provides explicit potential values for the solute’s infinite-dilution fugacity coefficient as determined by the Peng-Robinson (ECM-PR) and the Redlich-Kwong (ECM-RK). The ECM-PR, which is the Peng-Robinson version of this model, has been employed in this study. The following expression provides the fugacity coefficient in relation to the two adjustable parameters α and β:

$$\ln \phi _{2}^{\infty }=\beta \left( {Z - 1} \right) - \ln \left( {Z - B} \right)+\frac{A}{B}\left( {\beta - 2\alpha } \right)\ln \frac{{Z+B}}{Z}$$

(6)

Where, $\:\varnothing\:$ = fugacity coefficient of component, $\:Z$ = mixture compressibility factor, $\:R$ = universal gas constant, $\:T$ = temperature,$\:\:P$ = pressure, $\:A$ = dimensionless attraction parameter, $\:B$ = dimensionless co-volume parameter, $\:\alpha\:=$ temperature-dependent attraction correction factor and $\:\beta\:$= adjustable interaction parameter of the ECM-PR model.

The experimental solubility data can be used to find the parameters α and β by regression analysis. at a given temperature whenever experimental solubility data are available at that temperature for a given system.

Expanded liquid model (UNIQUAC)

The fundamental, is fugacity condition for the equilibrium between the solid solute (2) and SCCO₂ solvent (1) can be represented as follows:

$$f_{2}^{{L,scC{O_2}}}=\,f_{2}^{{S,solute}}$$

(7)

The fugacity of the solid phase can be estimated by comparing it to the fugacity of the pure solute, assuming that the solubility of SCCO₂ in the solid solute is negligible. By contrast, when the system reaches equilibrium, the solute’s fugacity in both phases is same, matching that of the pure solute. The following equation characterizes the equilibrium in the extended liquid model, which treats the SCF phase as a liquid:

$$f_{2}^{{L,scC{O_2}}}=\,{\gamma _2}{y_2}f_{2}^{{0L}}$$

(8)

Under these conditions, the solubility, y₂, can be expressed as follows:

$$f_{2}^{{L,scC{O_2}}}=\,f_{2}^{{0S}}={\gamma _2}{y_2}f_{2}^{{0L}}$$

(9)

Prausnitz et al.⁵³. have established the association between $f_{2}^{{oL}}$ and $f_{2}^{{oS}}$:

$$\ln \left( {\frac{{f_{2}^{{0S}}}}{{f_{2}^{{0L}}}}} \right)=\frac{{ - \Delta H_{2}^{f}}}{R}\left( {\frac{1}{T} - \frac{1}{{{T_m}}}} \right) - \frac{{\Delta {c_p}}}{{RT}}\left( {\frac{{T - {T_m}}}{T}} \right)+\frac{{\Delta {c_p}}}{R}\ln \left( {\frac{T}{{{T_m}}}} \right)$$

(10)

We can express the equilibrium solubility as follows, ignoring variations in the solute’s heat capacity with temperature and assuming that the solid solute is infinitely diluted in the SCCO₂:

$${y_2}=\frac{1}{{\gamma _{2}^{\infty }}}\exp \,\left[ {\frac{{\Delta H_{2}^{f}}}{R}\left( {\frac{1}{{{T_m}}} - \frac{1}{T}} \right)} \right]$$

(11)

Here, $\:{T}_{m}$​ is melting temperature of component 2 (K), $\:{\varDelta\:H}_{2}^{f}$=fusion enthalpy of component 2 (J/mol), and.

the symbol $\gamma _{2}^{\infty }$ represents the solute’s activity coefficient when diluted indefinitely. For this project, $\gamma _{2}^{\infty }$ was computed using the UNIQUAC model, which examines two contributions: one combinatorial, which accounts for the main entropic contribution, $\gamma _{2}^{{c,\infty }}$, and a residual part, $\gamma _{2}^{{R,\infty }}$, which stands for the intermolecular forces that add up to the mixing enthalpy.

$$\ln \gamma _{2}^{\infty }=\ln \gamma _{2}^{{c,\infty }}+\ln \gamma _{2}^{{R,\infty }}$$

(12)

In this context, the molecules in the system, both in terms of structure and composition, affect chlordiazepoxide’s solubility:

$$\ln \gamma _{2}^{{c,\infty }}=1 - \frac{{{r_2}}}{{{r_1}}}+\ln \frac{{{r_2}}}{{{r_1}}} - {q_2}\frac{z}{2}\left( {1 - \frac{{{r_2}{q_1}}}{{{r_1}{q_2}}}+\ln \frac{{{r_2}{q_1}}}{{{r_1}{q_2}}}} \right)$$

(13)

In this relationship, z commonly thought of as being equal to 10, is the coordination number. In this context, q is a (dimensionless) surface area measurement and r is the volume parameter. By the way, the parameter $\gamma _{2}^{{R,\infty }}$ is solely dependent on the intermolecular forces, as expressed in the following relationship:

$$\ln \,\gamma _{2}^{{R,\infty }}={q_2}\frac{{\alpha _{{12}}^{\prime }}}{{{T_r}}}+{q_2}\left( {1 - {e^{ - \frac{{\alpha _{{21}}^{\prime }}}{{{T_r}}}}}} \right)$$

(14)

where, the parameters $\alpha _{{12}}^{\prime }$ and $\alpha _{{21}}^{\prime }$ stand for the solute- SCCO₂ interaction energy, which is shown by the binary interaction parameters. Hence, they are expressed as functions of density, as indicated below:

$$\alpha _{{12}}^{\prime }={\alpha _{12}}{\kern 1pt} \rho _{r}^{{{\beta _{12}}}}$$

(15)

$$\alpha _{{21}}^{\prime }=\alpha {}_{{21}}{\kern 1pt} \rho _{r}^{{{\beta _{21}}}}$$

(16)

These parameters are indeed influenced by temperature and pressure, especially at high pressures, and are not constant.

Perturbed-Chain statistical associating fluid theory (PC-SAFT)

The PC-SAFT EoS is a highly accurate thermodynamic model for predicting phase behavior of associating and chain-forming fluids. It decomposes the residual Helmholtz free energy into distinct physical contributions:

$$\:{\text{A}}^{\text{r}\text{e}\text{s}}=\:{\text{A}}^{\text{h}\text{s}}+\:{\text{A}}^{\text{c}\text{h}\text{a}\text{i}\text{n}}+\:{\text{A}}^{\text{d}\text{i}\text{s}\text{p}}+\:{\text{A}}^{\text{a}\text{s}\text{s}\text{o}\text{c}}$$

(17)

Where; $\:{A}^{hs}$ is the hard-sphere contribution,$\:{\:A}^{chain}$ is the chain (connectivity) contribution, $\:{A}^{disp}$ is the dispersion (van der Waals) contribution and $\:{A}^{assoc}$ is the association (hydrogen bonding) contribution. The full set of equations and the detailed theoretical background of the PC-SAFT model have been comprehensively described in our previous publications.

Results and discussion

Experimental evaluation

As mentioned in Sect. "Laboratory section", this work is aimed at quantifying the solubility of chlordiazepoxide in SCCO₂ under different process conditions. The experimental evaluation utilized seven evenly spaced pressures ranging from 12 MPa to 30 MPa and four equally spaced temperatures ranging from 308 K to 338 K. for a total of 28 data points. Data points were calculated as the average of three repeated measurements to ensure accuracy. Table 5 presents the average solubility resulting from three measurements at each pressure-temperature combination for the chlordiazepoxide-SCCO₂ system. The solubility values measured should be carefully considered, as they correspond to a solid-supercritical fluid (solid-vapor) equilibrium. This is because chlordiazepoxide remained in the solid phase, while CO₂ was in its supercritical state, under all experimental conditions.

Table 5 Measured solubility (in PPm and mg/L) of chlordiazepoxide in SCCO₂.

The density of SCCO₂, an important parameter in modeling, is shown in Table 6, which came from the NIST database.

Table 6 Density of SCCO₂ within the studied range of temperature and pressure.

See how (a) SCCO₂ density and (b) pressure relate to the data in Table 5 in Fig. 3. The hydrolysis of chlordiazepoxide in SCCO₂ is affected by changes in temperature and pressure, as indicated by these results. Research on the solubility of chlordiazepoxide was conducted to increase with pressure under isothermal conditions, in agreement with the expected behavior. The observed rise in solubility is due to the reduced intermolecular distance between CO₂ and solute molecules; this it is a well-documented behavior^1,31. The closer proximity of CO₂ molecules creates a denser supercritical CO₂ phase, enhancing the interactions between chlordiazepoxide and CO₂ molecules. This strengthened interaction ultimately leads to improved drug solubility in the supercritical phase.

Fig. 3

Experimental solubility of chlordiazepoxide in SCCO₂: (a) solubility vs. density, and (b) solubility vs. pressure.

The isotherms different slopes and the crossover point somewhere between 18 and 20 MPa show that they follow different trends, as seen in Fig. 3(b). The polar opposite effects of temperature on two solubility-influencing variables are responsible for this result, i.e., solute sublimation pressure and SCCO₂ density^{45,54,55,56,57,58}. As mentioned above, it appears that at a pressure of about 19 MPa, the four isotherms cross each other. This means that beyond this pressure threshold, an increase in solubility is observed with temperature, while below the crossover pressure, the solubility decreases at higher temperatures. This is because sublimation pressure, density, and solubility are all affected by temperature in inverse proportions, which explains the observed phenomenon. Density increases the solvation power, which implies that solubility reduces as temperature drops. Because it grows exponentially with temperature, sublimation pressure becomes the most important factor at higher temperatures) and therefore the solubility is higher at higher temperatures.

To provide clearer context for the present findings and to further demonstrate both their validity and originality, a comprehensive comparative table has been compiled and included in the Supplementary Information. This table summarizes key solubility data reported in the literature for a wide range of pharmaceutical compounds in supercritical CO₂, encompassing both studies conducted by other researchers and the authors’ earlier works. The consistency between the solubility trend observed for chlordiazepoxide and the reported ranges of similar drug–CO₂ systems supports the reliability and robustness of the current results.

Theoretical evaluation

Empirical models

To ensure a leveled playing field when comparing the four empirical models included in this study (see Table 2), each model has three changeable parameters. The model parameters that can be adjusted were found using optimization algorithms to minimize AARD%. Alternatively, R² could have been maximized to find the parameters a₀, a₁, and a₂, but experience has shown that the difference in the results is not significant. The results are summarized in Table 7, where R² are also displayed.

Table 7 Correlation results of chlordiazepoxide solubility data in SCCO₂.

Referring to Tables 2 and 7; Fig. 4 displays the outcomes of the correlations with the experimental data for all four empirical models. Kumar-Johnson (K-J)⁵⁹, Bartle⁶⁰, Chrastil⁶¹, and Méndez-Santiago and Teja (MST)⁴¹ are the ones listed from top left to bottom right. All models demonstrate dependable accuracy in correlating the data for this binary system, as shown in Table 7; Fig. 4. Oddly enough, when looking at AARD% and R², the Chrastil model, the oldest of the four models that were tested produces the best results.

Fig. 4

Comparison of the experimental and the empirical model-estimated solubility data for chlordiazepoxide- SCCO₂ system.

Equation of state-based model: SRK

The solubility behavior of chlordiazepoxide in SCCO₂ was successfully correlated using the Soave-Redlich-Kwong (SRK) equation of state. As illustrated in Fig. 5, the SRK model is capable of reproducing the general trend of the experimental solubility data across all four investigated temperatures. Although some deviations are visible at high pressures, where the experimental solubility increases more steeply, the overall shape and pressure-dependency of the solubility curves are satisfactorily predicted. The numerical performance indicators summarized in Table 8 further support this observation. The correlation coefficient (R²) remains consistently high for all temperatures, with an average value of 0.963, indicating that the model captures most of the variability in the experimental data. The AARD% across the full dataset is approximately 20%, which, although not negligible, is acceptable considering the complexity of solute-supercritical solvent interactions and the sensitivity of SCCO₂systems to small changes in pressure and temperature. Despite the moderate AARD% values, the graphical comparison reveals that the SRK model provides a reasonable representation of the solubility profile, especially in the mid-pressure region where experimental and calculated data nearly overlap. This agreement suggests that the SRK model, while relatively simple, is capable of capturing the essential thermodynamic behavior of the chlordiazepoxide + SCCO₂ system. Overall, the results highlight that SRK delivers a reliable first-order prediction of solubility trends, even if some refinements would be required for high-precision applications.

The solubility of chlordiazepoxide in SCCO₂ is accurately depicted in Fig. 5, which uses the SRK model. Table 8 also shows the AARD%, R², and ideal temperature-dependent SRK model parameters. The SRK model’s output values are well correlated with the empirical data. The average values of these parameters (considering all four temperatures) are AARD% = 20.04, and R² = 0.9631. Despite these not-so-good values, the graphical comparison of the calculated solubility data by the SRK model and the corresponding experimental data in Fig. 5 show a reasonable agreement.

Table 8 Results of the SRK model for the chlordiazepoxide solubility in SCCO₂.

Fig. 5

Solubilities for the chlordiazepoxide in SCCO₂: Experimental values and SRK-model.

Equation of state-based model: ECM-PR

In this case, we will employ the EoS model that Estévez et al.⁵⁵ suggested. An intriguing aspect of this model, in contrast to previous EoS models, is that it can function without the solute’s crucial qualities, as they are incorporated into two adjustable parameters α and β. This method is applicable to every EoS. The solute’s infinite-dilution fugacity coefficient is explicitly expressed in the original publication⁵⁵ using the Peng-Robinson EoS (ECM-PR) and the Redlich-Kwong EoS (ECM-RK). The ECM-PR, which is the Peng-Robinson version of this model, has been employed in this study. The fugacity coefficient, expressed as a function of the two adjustable parameters α and β, is given by Eq. (6) in the ECM-PR Model section.

The experimental solubility data can be used to find the parameters α and β by regression analysis. at a given temperature whenever experimental solubility data are available at that temperature for a given system. In the case of chlordiazepoxide, the experimental data used are those reported in Table 5 at 35 °C, 45 °C, 55 °C, and 65 °C. Several statistical criteria can be used in the regression of the experimental data to obtain α and β at any given temperature. In this work, AARD was chosen, for the sake of illustration, since it is probably among the most common ones.

Figure 6 presents a graphical summary of the calculation results in which the solubility is represented, at the four temperatures studied, both in relation to (a) the density of SCCO₂ and (b) the pressure.

Fig. 6

Experimental solubility (symbols) and ECM-PR model solubility (lines) of chlordiazepoxide in SCCO₂: (a) as a function of density, and (b) as a function of pressure.

Note that the log scale exaggerates the differences between model and experimental values in the low-end range. Table 9 shows the resulting values of α and β as well as statistical parameters of the fit all four temperatures.

Table 9 Results of the ECM-PR model for the chlordiazepoxide solubility in SCCO₂.

Interestingly, the crossover pressure mentioned in the context of the discussion of the experimental data (Fig. 3, above) is confirmed by the model. As observed, the crossover occurs somewhere between 17 MPa and 19 MPa. It should be emphasized that EoS models, in general, are more reliable when extrapolation of the solubility outside the pressure range used in the experimentation is needed. This is true at higher or lower pressures of those used in the lab, but particularly at lower ones as shown in the graphical abstract of reference⁵⁵.

Expanded liquid model (UNIQUAC)

Given the satisfactory precision demonstrated by the UNIQUAC model in correlating solubility data of different materials in supercritical fluids, it has been used here for the solubility of chlordiazepoxide in SCCO₂. The parameters q and r for chlordiazepoxide (q₂ and r₂), were determined using a group contribution method while the parameters for CO₂ (q₁ and r₁) were obtained from references⁴⁰. Table 10 summarizes these values.

Table 10 UNIQUAC pure species parameters.

Table 11 provides the parameters of this model derived from the experimental data by regression, in addition to the overall statistical parameters of AARD%, and R₂, (overall in the sense that is obtained combining the results at the four temperatures). The outcomes of chlordiazepoxide solubility correlation with the UNIQUAC model are shown in Fig. 7. As observed, the UNIQUAC model accurately models the supercritical solubility data of chlordiazepoxide.

Table 11 UNIQUAC binary and overall statistical parameters for the studied system.

Fig. 7

Experimental and UNIQUAC-estimated solubility of chlordiazepoxide in SCCO₂.

PC-SAFT

As outlined in Table 12, the segment number (m), segment energy (ε/k), and segment diameter (σ) are reported for carbon dioxide and the pharmaceutical agent. The PC-SAFT parameters for CO₂ were adopted from the literature³⁸, whereas the drug parameters (m, σ, and ε/k) were obtained by regression against experimental solubility data. The PC-SAFT was applied to optimize the solubility model parameters for the drug in CO₂, and the resulting SAFT parameters are listed in Table 12. The mean AARD% value of 8.85, in conjunction with the R² of 0.9859 reported in Table 13. This conclusion is further substantiated by the close agreement between experimental and calculated solubility data, as illustrated in Fig. 8. Furthermore, as illustrated in Table 9, an inverse relationship is observed between temperature and the binary interaction parameter K₁₂.

Table 12 The parameters of PC-SAFT EoS.

Table 13 Results of the SAFT for the chlordiazepoxide solubility in SCCO₂.

Fig. 8

Experimental and PC-SAFT results solubility of chlordiazepoxide in SCCO₂.

Model performance comparison

To provide a clearer assessment of the predictive capabilities of the different thermodynamic models, a comprehensive numerical and graphical comparison was carried out between the empirical models, the EoS models, and the expanded-liquid UNIQUAC model. As summarized in the comparative table (Table 14), the empirical correlations, particularly the Chrastil model exhibited the highest agreement with the experimental solubility data, yielding an AARD% of 5.30 and an R² value of 0.996. The MST and Bartle correlations also performed satisfactorily, with deviations below 7%. In contrast, the EoS-based models (SRK and ECM-PR) showed significantly higher deviations, with average AARD% values close to 20%, indicating their lower accuracy for this system when used without extensive parameter tuning. The UNIQUAC model demonstrated strong predictive behavior (AARD% = 6.12; R² = 0.988), performing comparably to the best empirical correlations. The graphical comparison of AARD% values (Fig. 9) similarly highlight the superior performance of the Chrastil model followed by UNIQUAC. Overall, these results indicate that density-dependent empirical correlations and activity-coefficient–based approaches provide a more accurate representation of the solubility behavior of chlordiazepoxide in SCCO₂ than cubic EoS models.

Fig. 9

Comparison between experimental solubility data of chlordiazepoxide in SCCO₂ and the predictions obtained from the applied thermodynamic models.

Table 14 Comparative performance of the applied thermodynamic models against experimental solubility data of chlordiazepoxide in SCCO₂.

Conclusion

To create a novel pharmaceutical formulation of chlordiazepoxide with enhanced effectiveness using a supercritical method, it is indeed crucial to obtain its supercritical solubility data. Since no prior data on this system appear to exist, the primary goal of this investigation was to determine the solubility of chlordiazepoxide in SCCO₂ under various conditions. The range of temperatures investigated was from 308 K to 338 K, with seven equidistant pressures covering 12 MPa to 30 MPa. From 19.8 to 576 ppm on a mole basis, the drug’s solubility was determined to be between 0.0198 × 10 ^− 3 and 0.576 × 10 ^–3 mol fractions. To theoretically analyze the supercritical solubility of chlordiazepoxide in SCCO₂, several empirical models were employed. With an AARD% of approximately 5.3%, and R² of 0.996, the empirical models created by Chrastil for this system performed exceptionally well. The experimental findings and the solubility values predicted by the thermodynamic models were very congruent. In particular, when comparing the UNIQUAC model to the SRK model, the UNIQUAC model showed better accuracy in forecasting chlordiazepoxide solubility in SCCO₂, with an AARD% of 6.12% and a R² of 0.981. This indicates that the UNIQUAC model is more reliable in estimating the solubility behavior of chlordiazepoxide in SCCO₂compared to the SRK model.